# How do you find the equation for a hyperbola centered at the origin with a vertical transverse axis of lengths 12 units and a conjugate axis of length 4 units?

Dec 15, 2017

The general form for a the equation of a hyperbola with a vertical transverse axis is:

${\left(y - k\right)}^{2} / {a}^{2} - {\left(x - h\right)}^{2} / {b}^{2} = 1$

#### Explanation:

The center of the general form is $\left(h , k\right)$. We are told that the center is the origin: $\left(0 , 0\right)$, therefore, we can remove h and k from the equation:

${y}^{2} / {a}^{2} - {x}^{2} / {b}^{2} = 1$

The length of the transverse axis is $2 a$

$2 a = 12$

$a = 6$

${y}^{2} / {6}^{2} - {x}^{2} / {b}^{2} = 1$

The length of the conjugate axis is $2 b$:

$2 b = 4$

$b = 2$

${y}^{2} / {6}^{2} - {x}^{2} / {2}^{2} = 1$

Done.